Pointwise limit of increasing sequence of cadlag (or RCLL) functions.

Question

Is the pointwise limit of non-decreasing càdlàg functions càdlàg? if not, is possible to add hypothesis in order to get limit function cadlag?

Answer 1

Consider :

$$f_n(x)=\left\lbrace \begin{matrix} 0 && \text{if } x <0,\\ 1 && \text{if } x \in [0,1/n] ,\\ 2 && \text{if } x>1/n. \end{matrix} \right.$$

Each $f_n$ is RCLL, but they converge to :

$$f(x)=\left\lbrace \begin{matrix} 0 && \text{if } x <0,\\ 1 && \text{if } x=0 ,\\ 2 && \text{if } x>0. \end{matrix} \right.$$

Because of $x=0$, $f$ is not RCLL.

Since a sequence of increasing functions is an increasing function (even with pointwise convergence only), and that an increasing function always have a left limit at any point, the problem is the right continuous part.

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A sufficient condition would be the following.

Let $D(g)$ the set of discontinuities of $g$. That set is countable for increasing functions.

If $\exists N, \forall n \ge N, D(f_n)=D(f_N)=\{y^N_m\}_m$, with $\{y^N_m\}_m$ an increasing sequence and you have uniform convergence of $f_n$ on a interval of the form $[y^M_n,y]$ for some $y > y^M_n$, then the result follows.

PS : I'm not sure this is a necessary condition.